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Bose-Einstein Condensation
(For the 9th Edition of the McGraw-Hill Encyclopedia of Science & Technology)
Wolfgang Ketterle
Massachusetts Institute of Technology
Cambridge, MA 02139, USA
When a gas of bosonic particles is cooled below a critical temperature, it condenses into a
Bose-Einstein condensate. The condensate consists of a macroscopic number of particles, which
are all in the ground state of the system. Bose-Einstein condensation (BEC) is a phase-transition,
which does not depend on the specific interactions between particles. It is based on the
indistinguishability and wave nature of particles, both of which are at the heart of quantum
mechanics.
Basic phenomenon in an ideal gas
In a simplified picture, particles in a gas may be regarded as quantum-mechanical wavepackets
which have an extent on the order of a thermal de Broglie wavelength λdB = (2π=2/mkBT)1/2
where T is the temperature, m the mass of the particle, kB Boltzmann’s constant and = Planck’s
constant. λdB can be regarded as the position uncertainty associated with the thermal momentum
distribution. At high temperature, λdB is small, and it is very improbable to find two particles
within this distance. Therefore, the indistinguishability of particles is not important, and a
classical description applies (Boltzmann statistics). When the gas is cooled to the point where
λdB is comparable to the distance between particles, the individual wavepackets start to “overlap”
and the indistinguishability of particles becomes crucial – an “identity crisis” occurs. For
fermions, the Pauli exclusion principle prevents two particles from occupying the same quantum
state, whereas for bosons, quantum statistics dramatically increases the probability of finding
several particles in the same quantum state (Bose-Einstein statistics). The system undergoes a
phase transition and forms a Bose-Einstein condensate, where a macroscopic number of particles
occupy the lowest-energy quantum state (Fig. 1). The temperature and the density of particles n
at the phase transition are related as nλdB3= 2.612. BEC is a phase-transition solely caused by
quantum statistics, in contrast to other phase-transitions (like melting or crystallization) which
depend on the inter-particle interactions.
Fig. 1: Criterion for Bose-Einstein condensation. At
high temperatures, a weakly interacting gas can be
treated as a system of “billiard balls.” In a simplified
quantum description, the particles can be regarded as
wavepackets with an extension λdB. At the BEC
transition temperature, λdB becomes comparable to the
distance d between particles, and a Bose-Einstein
condensate forms (in the case of bosonic particles). As
the temperature approaches zero, the thermal cloud
disappears leaving a pure Bose condensate.
Bose-Einstein condensation can be described intuitively in the following way: when the
quantum-mechanical wavefunctions of bosonic particles spatially “overlap”, then the matter
waves start to oscillate in concert - a coherent matter wave forms that comprises all particles in
the ground state of the system. This transition from “disordered” to coherent matter waves can
be compared to the step from incoherent light to laser light. Indeed, atom lasers based on BoseEinstein condensation have been recently realized.
Bose-Einstein condensation occurs in thermal equilibrium. Although the condensate itself
does not contribute to the entropy, its presence maximizes the entropy of the other particles,
which are distributed among the higher energy states.
Atoms and molecules are composite particles. They are bosonic (fermionic) if they have
integer (half-integer) spin, or equivalently, if the total number of electrons, protons, and neutrons
they contain is even (odd). As long as the thermal energy is much less than internal excitation
energies of the composite particles, we can regard them as pointlike particles obeying BoseEinstein (Fermi-Dirac) statistics.
Experimental techniques
The phenomenon of Bose-Einstein condensation is responsible for the superfluidity of helium
and for the superconductivity of an electron gas which involves “Bose condensed” electron pairs.
However, these phenomena happen at high density and require a detailed treatment of the
interactions.
The quest to realize BEC in a dilute weakly interacting gas was pursued in several different
directions: liquid helium diluted in vycor, a sponge-like glass, excitons in semiconductors, which
consist of weakly-bound electron-hole pairs, and in atomic gases. This article discusses the last
case since most of the research activities have focused on these systems.
At ultralow temperatures, all atomic gases liquefy or solidify in thermal equilibrium. Keeping
the gas at sufficiently low density can prevent this. Elastic collisions, which ensure thermal
equilibrium of the gas, are then much more frequent than three-body collisions, which are
required for the formation of molecules and other aggregates. As a result the gaseous phase is
metastable for seconds or even minutes and allows for the observation of BEC in a gas. Typical
atomic densities between 1012 and 1015 particles/cm3 imply BEC transition temperatures in the
nanokelvin or microkelvin regime.
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Table 1: Multi-stage cooling to BEC in the MIT sodium experiment. Through a combination of laser and
evaporative cooling, the temperature of the gas is reduced by a factor of 109, while the density at the BEC transition
is similar to the initial density in the atomic beam oven (all numbers are approximate). In each step shown, the
ground state population (which is proportional to the phase-space density) increases by about 106, resulting in a total
increase of phase-space density by 20 orders of magnitude.
Oven
Laser cooling
Evaporative cooling
BEC
Density [cm-3]
1014
1011
1014
1015
Temperature
500 K
50 µK
500 nK
Phase-space density
10-13
10-6
2.612
107
The realization of BEC in atomic gases required techniques to cool gases to such low
temperatures, and atom traps to confine the gas at the required density and keep them away from
the much warmer walls of the vacuum chamber. The experiments on alkali vapors (lithium,
rubidium, sodium) use several laser-cooling techniques as precooling, then hold the atoms in a
magnetic trap and cool them further by forced evaporative cooling (Fig. 2 and Table 1). For
atomic hydrogen, the laser-cooling step is replaced by cryogenic cooling. Evaporative cooling is
done by continuously removing the high-energy tail of the thermal distribution from the trap.
The evaporated atoms carry away more than the average energy, which means that the
temperature of the remaining atoms decreases. The high-energy tail must be constantly
repopulated by collisions, thus maintaining thermal equilibrium and sustaining the cooling
process. Evaporative cooling is a common phenomenon in daily life – it’s how hot water cools
down in a bathtub or in a cup of coffee.
Fig 2: Experimental setup for cooling a gas of
atoms to Bose-Einstein condensation. A cold cloud
of atoms is confined in a magnetic trap and cooled
by evaporative cooling. Flipping their spin with RF
radiation evaporates the highest energy particles.
Since this process reverses the magnetic force, these
particles are expelled from the trapping region.
Observation of Bose-Einstein condensation
Bose-Einstein condensation was observed by suddenly switching off the atom trap releasing
the atom cloud into a ballistic expansion. The cloud was illuminated with resonant laser light,
and its shadow was imaged onto a CCD camera (Fig. 3). Such images showed the sudden
appearance of a bimodal cloud consisting of a diffuse normal component and a dense core (the
condensate) near the predicted transition temperature. The condensate expanded much faster
than corresponding to the zero-point energy of the trapping potential – clear evidence for
repulsive interactions between the atoms, which accelerate the ballistic expansion.
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Fig. 3: Observation of Bose-Einstein condensation by
absorption imaging. Shown is absorption vs. two
spatial dimensions. The Bose-Einstein condensate is
characterized by its slow expansion observed after 6
msec time of flight. The left picture shows an
expanding cloud cooled to just above the transition
point; middle: just after the condensate appeared; right:
after further evaporative cooling has left an almost pure
condensate. The total number of sodium atoms at the
phase transition is about 7x105; the temperature Tc at
the transition point is 2 µK.
The sudden reduction of the speed of ballistic expansion has been called condensation in
momentum space. However, BEC is a condensation into the lowest energy state; in an
inhomogeneous trapping potential, the lowest state has the smallest dimension. Condensation
into this states thus results in the formation of a dense core of Bose condensed atoms within a
diffuse thermal cloud of atoms. This spatial structure of a Bose condensate was directly observed
using dispersive imaging techniques.
Weak interactions
Bose-Einstein condensation in gases allows for a “first-principles” theoretical description
because there is a clear hierarchy of length and energy scales. In a gas, the separation between
atoms n-1/3 is much larger than the effective range of the interatomic forces (characterized by the
s-wave scattering length a, typically 1 to 5 nm for alkali atoms), i.e. the quantity na3<<1. This
inequality expresses that binary collisions are much more frequent than three-body collisions. It
is in this limit that the theory of the weakly interacting Bose gas applies. Interactions between
the Bose condensed are described by a mean field energy Uint = 4 π =2 n a /m. The stability of
large condensates requires repulsive interactions (positive a). For negative a, the interactions are
attractive, and the condensate is unstable against collapse above a certain size.
Properties of the weakly interacting Bose gas have been studied both experimentally and
theoretically. Of special interest were the collective excitations of the gas, different forms of
sound (Bogoliubov, first and second sound) and the stability of vortices.
When condensates were realized in a mixture of atomic hyperfine states, multi-component
condensates were observed. Their phase diagram, the spatial structure, possible metastable
configurations and properties like miscibility and immiscibility are examples of new phenomena
which have been studied. These examples show that gaseous Bose condensates are a new class
of quantum fluids with properties quite different from the quantum liquids 3He and 4He. They
provide a testing ground for aspects of many-body theories, which were developed many decades
ago but never tested experimentally.
Macroscopic wavefunction
In an ideal
wavefunction.
single-particle
single-particle
wavefunction.
gas, Bose condensed atoms all occupy the same single-particle ground-state
The many-body ground-state wavefunction is then the product of N identical
ground-state wavefunctions where N is the number of condensed atoms. This
wavefunction is therefore called the condensate wavefunction or macroscopic
This picture is valid even when weak interactions are included. The ground state
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many-body wavefunction is still, to a very good approximation, a product of N single-particle
wavefunctions, which are now obtained from the solution of a non-linear Schrödinger equation,
which includes the interaction energy between atoms. The admixture of other configurations into
the ground state is called quantum depletion. The theory of weakly interacting Bose gases shows
that the fractional quantum depletion is (8 / 3π 1 / 2 ) na 3 , typically 1 % or less for the alkali
condensates. This means that even for the interacting gases, with 99 % accuracy, all the atoms
can be regarded to have the same single-particle wavefunction. This is in contrast to liquid
helium in which the quantum depletion is about 90 %.
The phase of the condensate and the coherence properties of its wavefunction are relevant for
quantum fluids because the gradient of the phase is proportional to the superfluid velocity. It
also determines the properties of atom lasers based on BEC. In superconductors and liquid
helium, the existence of coherence and of a macroscopic wavefunction is impressively
demonstrated through the Josephson effect. In the dilute atomic gases, the coherence has been
demonstrated even more directly by interfering two Bose condensates (Fig. 4).
Fig. 4: Interference pattern of two expanding condensates
demonstrating the coherence of Bose-Einstein condensates.
This absorption image was observed after 40 ms time of
flight. The interference fringes have a spacing of 15 µm, a
huge length for matter waves. In contrast, the matter
wavelength of atoms at room temperature is only 0.5 Å, less
than the size of an atom.
For background information see Bose-Einstein statistics, Laser cooling, Particle trap, De Broglie
wavelength, Atom laser, Coherence.
Bibliography
1.
E.A. Cornell, C.E. Wieman, The Bose-Einstein Condensate, Scientific American, March 1998, pp. 40 –45.
2.
M. Inguscio, S. Stringari, and C.E. Wieman (eds.), Bose-Einstein condensation in atomic gases, Proceedings of
the International School of Physics “Enrico Fermi”, in print (1999).
3.
A. Griffin, D.W. Snoke, and S. Stringari (eds.), Bose-Einstein condensation (University Press, Cambridge,
1995).
4.
D.S. Durfee and W. Ketterle, Experimental studies of Bose-Einstein condensation, Optics Express 2, 299-313
(1998). See: http://epubs.osa.org/opticsexpress.
5.
BEC home page of the Georgia Southern University: http://amo.phy.gasou.edu/bec.html.
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